In the two dimensional real vector space ${ℝ}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $ℱ$ of such number systems. This is the set of all elements of ${ℝ}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $ℱ$ is a compact set and certain translates of it form a tiling of the ${ℝ}^2$. We construct points, where...

Each homeomorphism from the n-dimensional Sierpiński gasket into itself is a similarity map with respect to the usual metrization. Moreover, the topology of this space determines a kind of Haar measure and a canonical metric. We study spaces with similar properties. It turns out that in many cases, "fractal structure" is not a metric but a topological phenomenon.

We formulate variants of the cardinals f, p and t in terms of families of measurable functions, in order to examine the effect upon these cardinals of adding one random real.

We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that-depending on the function space and the set Γ-there occurs an...

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that consistently there exists an uncountable coanalytic subset of the plane that intersects every C¹ curve in a countable set.

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R 1 resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution...

Let X and Y be two compact spaces endowed with respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on X x Y whose marginals coincide with μ and ν, and such that the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first show that if the cost function c is decomposable, i.e., can be represented as the sum of two continuous functions defined on X and Y, respectively,...

A properly measurable set $𝒫\subset X\times M_1\left(Y\right)$ (where $X,Y$ are Polish spaces and $M_1\left(Y\right)$ is the space of Borel probability measures on $Y$) is considered. Given a probability distribution $\lambda \in M_1\left(X\right)$ the paper treats the problem of the existence of $X\times Y$-valued random vector $(\xi ,\eta )$ for which $ℒ\left(\xi \right)=\lambda$ and $ℒ\left(\eta \right|\xi =x)\in {𝒫}_x$$\lambda$-almost surely that possesses moreover some other properties such as “$ℒ(\xi ,\eta )$ has the maximal possible support” or “$ℒ\left(\eta \right|\xi =x)$’s are extremal...

Let $\mathrm{SP}$ be the set of upper strongly porous at $0$ subsets of ${ℝ}^{+}$ and let $\widehat{I}\left(\mathrm{SP}\right)$ be the intersection of maximal ideals $I\subseteq \mathrm{SP}$. Some characteristic properties of sets $E\in \widehat{I}\left(\mathrm{SP}\right)$ are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of ${ℝ}^{+}$ is a proper subideal of $\widehat{I}\left(\mathrm{SP}\right).$ Earlier, completely strongly porous sets and some of their properties were...

A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some $\sigma$-ideal, being (completely) nonmeasurable with respect to different $\sigma$-ideals, being a $\kappa$-covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations...

Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?